Preprint 2006-021

Stefano Bianchini and Laura V. Spinolo

Abstract:
We study the limit of the hyperbolic-parabolic approximation

vεt+A(vε,εvεx)vεx = εB(vε)vε_{xx},

with data
β(vε(t,0)) = g and
vε(0,x) = v0.
The function β is defined in such a way to guarantee
that the initial boundary value problem is well posed
even if B is not invertible.
The data g and v0
are constant.

When B is invertible,
the boundary datum can be assigned by imposing
vε(t,0) = vb,
where vb is again a constant.
The conservative case is included in the previous formulations.

It is assumed convergence of the
vε,
smallness of the total variation
and other technical hypotheses
and it is provided a complete characterization of the limit.

The most interesting points are the following two.

First, the boundary characteristic case is considered,
i.e. one eigenvalue of A can be 0.

Second, as pointed out before we take into account
the possibility that B is not invertible.
To deal with this case, we take as hypotheses
conditions that were introduced by Kawashima and Shizuta
relying on physically meaningful examples.
We also introduce a new condition of block linear degeneracy.
We prove that, if it is not satisfied,
then pathological behaviours may occur.

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